(a) Given z = x + jy determine if f (z) = z4 is analytic.(b) On an Argand Diagram sketch the region |z| < 1.(c) Map the region |z| < 1 into the function plane f (z) = U + jV , defined as f (z) = z4.
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(b) On an Argand Diagram sketch the region |z| < 1.
(c) Map the region |z| < 1 into the function plane f (z) = U + jV , defined as f (z) = z4.
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- /4-x² – y² 2. Let f(x,y)=: Shade the region in the x-y plane that corresponds to the domain of f (x, y) In(x- y)Find the global min and max of the provided function on the provided region R: f(x,y) = x - y - xy on the triangle R with vertices (0,0), (0,2), and (4,0)Let S = (D), where D = {(u, v) : u² + v² ≤ 1, u ≥ 0, v ≥ 0} and Þ (u, v) = (2u + 1, u – v, 3u + v). (a) Calculate the surface area of S. (Express numbers in exact form. Use symbolic notation and fractions where needed.) area (S) (b) Evaluate f (3x – 3y) dS. Hint: Use polar coordinates. (Express numbers in exact form. Use symbolic notation and fractions where needed.) I S √6T 2 Incorrect (3x - 3y) (3x − 3y) dS = 1 3
- Compute F. dr, where F(x, y) = oriented counterclockwise. (Use symbolic notation and fractions where needed.) [F C -2y 2x (x² + y²) ²³ (x² + y²) ² F. dr = and C is a circle of radius 4 with center at the origin1. Consider the function f(r,v)- (a) Find and sketch the domain of f(z, y). Shade the region corresponding to the domain of this function and indicate whether you are including or excluding the boundary of the domain by either using a dotted or solid line. (b) Sketch the level curve f(r, y) = 1.A) Find and describe the region inscribed by the lines (x-axis), (y = -x+2) and (x = 2y-4) under the mapping function (w=1/z)
- Consider the function f (x, y, z) = 2x² + 5y² + z². a) If f(x, y, z) is the profit made from selling x phones y computers, and z tablets interpret and compute f(0, 2, 2). b) Sketch and identify the traces to f(x, y, z) = 15. C nts) Let F =, use Green's Theorem to evaluate F. dr where the curve C' is the curve shown in the picture consisting of functions y=x² − 2x going from point (0,0) to the point (3,3) followed by the line segment from (3,3) to (0,0). (0,0) y=x (3,3) C y=x²-2xLet S be the graph of f(x, y) = x2 + y2 - 2x + y for(x, y) ∈ D = {(x, y) : x2 + y2 ≤ 4 and x + y ≥ 0}. Let C1 be the curve along the edge of the graph of fover the straight edge of D – starting in the secondquadrant and ending in the fourth quadrant of D • Let C2 be the curve along the edge of the graph of fover the circular edge of D – starting in the secondquadrant and ending in the fourth quadrant of D. Let F(x, y, z) = ⟨x, y, y z⟩ Compute R: C1 F¯ · dr¯.